Meetings

Click on the theme item for the meeting plan for that theme.
Click on the meeting item for references, exercises, and additional reading related to it.

Theme 1 : Computability Theory - 29 meetings

Meeting 01 : Mon, Jul 31, 08:00 am-08:50 am

Brief introduction to the course and the contents. Administrative Announcements. Grading policies. Languages vs Algorithmic Problems. Automata theory vs algorithms courses. Historical aspects of the idea of computation. Ruler and Compass problem. Hilbert's problem. Need of understanding computation in very formal terms.

Meeting 02 : Tue, Aug 01, 12:00 pm-12:50 pm

Problem of Searching the Treasure on the number line. Algorithms. Drawbacks. Diophantine problem. Formal model of computation. Post Systems, Mu-calculus, Lambda calculus. Turing machines. Formal Model of Turing machines.

Meeting 03 : Thu, Aug 03, 11:00 am-11:50 am

Configurations of TM model. Acceptance of a string by Turing machines. Decidable, semi-decidable and undecidable. Robustness of the Turing machine model. Tape reduction theorem. Robustness of the Turing machine model. Multitape Turing machines, multi-head Turing machines.

Meeting 04 : Fri, Aug 04, 10:00 am-10:50 am

Encoding of Turing machines as strings. Universal Turing machine.

Meeting 05 : Mon, Aug 07, 08:00 am-08:50 am

Decidable Languages, Semi-Decidable Languages. Connections and structure.

Meeting 06 : Tue, Aug 08, 12:00 pm-12:50 pm

Story of infinity. Review of cantors diagonalization (1895). Undecidability of the halting problem (1936).

Meeting 07 : Thu, Aug 10, 11:00 am-11:50 am

Undecidability of Membership problem. Informal notion of reductions. Are there other undecidable problems. Examples of decidable problems.

Meeting 08 : Fri, Aug 11, 10:00 am-10:50 am

Examples of undecidable problems. One tool does it all. Undecidability of MP_101, EMPTY, TOTAL, FIN.

Meeting 09 : Mon, Aug 14, 08:00 am-08:50 am

To Be Announced

Meeting 10 : Thu, Aug 17, 11:00 am-11:50 am

To Be Announced

Meeting 11 : Fri, Aug 18, 10:00 am-10:50 am

How do we prove that a language is not semi-decidable? HP-bar reduces to FIN.

Meeting 12 : Sat, Aug 19, 03:00 pm-04:15 pm

Proof of Rice's First Theorem. Applications. Rice's Second Theorem. Proof and Applications.

Meeting 13 : Mon, Aug 21, 08:00 am-08:50 am

SD-completeness. CoSD-completeness. Hardness. MP is SD-complete. HP is SD-complete.

Meeting 14 : Tue, Aug 22, 12:00 pm-12:50 pm

Axioms and Rules of inteference. Proof system. Completeness, soundness. Peano's axioms. Godel's incompeteness Theorem. Turing's proof of the incompletness.

Meeting 15 : Thu, Aug 24, 11:00 am-11:50 am

Proof of incompletenss theorem.

Meeting 16 : Fri, Aug 25, 10:00 am-10:50 am

Proof of incompletenss theorem.

Meeting 17 : Mon, Aug 28, 08:00 am-08:50 am

Quiz 1

Meeting 18 : Tue, Aug 29, 12:00 pm-12:50 pm

No lecture, yet to be compensated.

Meeting 19 : Thu, Aug 31, 11:00 am-11:50 am

Post's theorem.

Meeting 20 : Fri, Sep 01, 10:00 am-10:50 am

Post's theorem.

Meeting 21 : Mon, Sep 04, 08:00 am-08:50 am

MPXOR example. Oracle Turing machines. OTM for MPXOR. Definition of "semidecidable in A", "decidable in A".

Meeting 22 : Tue, Sep 05, 12:00 pm-12:50 pm

Turing reducibility. Connections to many-one reducibility. Sigma-i, Pi-i and Delta-i. Relation between these classes. Relativisation of proofs.

Meeting 23 : Thu, Sep 07, 11:00 am-11:50 am

Diagonalization is relativisable. MP^MP is complete for Sigma-2. Sigma-i cap Pi-i is exactly Delta-i. Separation of the levels of AH.

Meeting 24 : Sat, Sep 09, 10:00 am-10:50 am

Characterization of the Arithmetic Hierarchy. Applications. MP, REG, DEC, TOTAL, FIN.

Meeting 25 : Mon, Sep 11, 08:00 am-08:50 am

Proof of the characterization.

Meeting 26 : Tue, Sep 12, 12:00 pm-12:50 pm

Proof of the characterization. FIN is Sigma-2 complete.

Meeting 27 : Thu, Sep 14, 11:00 am-11:50 am

FIN^HP is Sigma-2 complete.

Meeting 28 : Fri, Sep 15, 10:00 am-10:50 am

COF is Sigma-3 complete.

Meeting 29 : Mon, Sep 18, 08:00 am-08:50 am

No lecture.

Theme 2 : Time Complexity Classes
- 15 meetings
- Meeting 30 : Tue, Sep 19, 12:00 pm-12:50 pm
- Meeting 31 : Thu, Sep 21, 11:00 am-11:50 am
- Meeting 32 : Mon, Sep 25, 08:00 am-08:50 am
- Meeting 33 : Tue, Sep 26, 12:00 pm-12:50 pm
- Meeting 34 : Tue, Sep 26, 06:00 pm-07:00 pm
- Meeting 35 : Fri, Sep 29, 10:00 am-10:50 am
- Meeting 36 : Mon, Oct 02, 08:00 am-08:50 am
- Meeting 37 : Tue, Oct 03, 12:00 pm-12:50 pm
- Meeting 38 : Thu, Oct 05, 11:00 am-11:50 am
- Meeting 39 : Fri, Oct 06, 10:00 am-10:50 am
- Meeting 40 : Sat, Oct 07, 11:00 am-11:50 am
- Meeting 41 : Mon, Oct 09, 08:00 am-08:50 am
- Meeting 42 : Tue, Oct 10, 12:00 pm-12:50 pm
- Meeting 43 : Thu, Oct 12, 11:00 am-11:50 am
- Meeting 44 : Fri, Oct 13, 10:00 am-10:50 am
Theme 3 : Space Complexity Classes
- 9 meetings
Meeting 45 : Mon, Oct 16, 08:00 am-08:50 am
To Be Announced
References
Exercises
Reading
To Be Announced
References : None
Meeting 46 : Tue, Oct 17, 12:00 pm-12:50 pm
To Be Announced
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To Be Announced
References : None
Meeting 47 : Thu, Oct 19, 11:00 am-11:50 am
To Be Announced
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To Be Announced
References : None
Meeting 48 : Fri, Oct 20, 10:00 am-10:50 am
To Be Announced
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To Be Announced
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Meeting 49 : Mon, Oct 23, 08:00 am-08:50 am
To Be Announced
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To Be Announced
References : None
Meeting 50 : Tue, Oct 24, 12:00 pm-12:50 pm
To Be Announced
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To Be Announced
References : None
Meeting 51 : Thu, Oct 26, 11:00 am-11:50 am
To Be Announced
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To Be Announced
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Meeting 52 : Fri, Oct 27, 10:00 am-10:50 am
To Be Announced
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To Be Announced
References : None
Meeting 53 : Mon, Oct 30, 08:00 am-08:50 am
To Be Announced
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To Be Announced
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Theme 4 : Randomized Complexity Classes
- 10 meetings
Meeting 54 : Tue, Oct 31, 12:00 pm-12:50 pm
To Be Announced
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Reading
To Be Announced
References : None
Meeting 55 : Thu, Nov 02, 11:00 am-11:50 am
To Be Announced
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To Be Announced
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Meeting 56 : Fri, Nov 03, 10:00 am-10:50 am
To Be Announced
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To Be Announced
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Meeting 57 : Mon, Nov 06, 08:00 am-08:50 am
To Be Announced
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To Be Announced
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Meeting 58 : Tue, Nov 07, 12:00 pm-12:50 pm
To Be Announced
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To Be Announced
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Meeting 59 : Thu, Nov 09, 11:00 am-11:50 am
To Be Announced
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To Be Announced
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Meeting 60 : Fri, Nov 10, 10:00 am-10:50 am
To Be Announced
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To Be Announced
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Meeting 61 : Mon, Nov 13, 08:00 am-08:50 am
To Be Announced
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To Be Announced
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Meeting 62 : Tue, Nov 14, 12:00 pm-12:50 pm
To Be Announced
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To Be Announced
References : None
Meeting 63 : Thu, Nov 16, 11:00 am-11:50 am
To Be Announced
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To Be Announced
References : None

CS6014 - Computability and Complexity

Today : Wed, May 15, 2024

Announcements

Meetings

Resource bounded computation. Blum's axiom. Examples and non-examples. Time, Space, Halt-Indicator. g-bounded complexity class. DTIME as a complexity class.

Borodin-Trankenbrot Gap theorem. Towards time hierarchy theorem. Weak, Weaker and Stronger versions. Discussions.

Proof of the hierarchy theorem. Attempt 1. Failure. Fixing to attempt 2, tape reduction theorem.

Optimality of tape reduction. The PALINDROME language. Hennie-Stearns theorem (statement) and discussion.

Crossing sequence argument and limitation of 1-tape Turing machines.

Notion of efficiency. Edmonds proposition. Discussion and the definition of class P, EXP. Basic simulations. Example problems - CLIQUE, REACH, INDSET, VC.

More examples - COMPOSITE, CIRCUITS, CKTSAT, CVP, MINCKT, ANTHAKSHARI. Exponential time trivial algorithms.

Graph Isomorphism. From the examples, the property of problems for "easily verifiable short certificates of membership". The class NP. Definition based on non-deterministic polytime Turing machines.

Non-deterministic Algorithms. Correctness and Runtime arguments. Equivalence of the two definitions of NP. NP is contained in EXP.

Padding arguments. NP=P implies NEXP=EXP. PRIMES and the class CoNP. CoNP is contained in EXP. P is contained in the intersection of NP and CoNP. Why complementation is a big deal.

References
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TAUT, EXACT-CLIQUE. If NP=P, which languages have polynomial time algrithms. Oracle Turing machines. Non-deterministic oracle Turing machines. Polynomial hierarchy.